Ultramicroscopy 42-44 (1992) 1509-1513
North-Holland
,mli~nefm;~v~n
A theoretical comparison between interferometric and optical
beam deflection technique for the measurement of cantilever
displacement in AFM
C o n s t a n t A.J. P u t m a n *, B a r t G. d e G r o o t h , N i e k F. van H u l s t a n d J a n G r e v e
Department of Applied Physics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
Received 12 August 1991
A shot-noise- and diffraction-limited Michelson interferometer and two optical beam deflection configurations are
compared for application in an atomic force microscope. The results show that under optimal conditions the optical beam
deflection method is just as sensitive as the interferometer. This remarkable result is explained by indicating the physical
equivalence of both methods.
1. Introduction
In the original set-up of the atomic force microscope of Binnig et al. [1], the cantilever displacement was measured by detecting changes in
tunneling current between a tunnel tip and the
backside of the cantilever. In the past few years a
variety of other methods to measure the cantilever displacement have been described including capacitance measurement [2,3] and various
optical methods. For a detailed analysis of the
sensitivity of the capacitance method we refer to
ref. [3]. The optical methods appear to be the
most used and can be divided into two categories:
interferometry [4-9] and optical beam deflection
(OBD) [10,11]. The OBD method is successfully
used in photothermal deflection spectroscopy; a
description of the physical background of this
technique can be found in ref. [12].
Intuitively one would expect that interferometry is more sensitive than optical beam deflection.
In practice, however, atomic resolution is obtained equally well with both systems, whereas an
optical beam deflectometer is much simpler to
* To w h o m correspondence should be, directed.
construct than an interferometer. This has led us
to take a closer look at the theoretical sensitivity
of both methods. We will show that in terms of
signal-to-noise ratio both principles yield very
similar results. In fact, from a physical point of
view, both methods are equivalent.
2. The sensitivity of an ideal Micheison interferometer
We will first find an expression for the signalto-noise ratio of an ideal interferometer. Fundamentally the minimal displacement that can be
measured with an interferometer is determined
by the wave nature of light and the Poisson
disitribution of the photon-emission rate of the
light source. Other sources of noise, such as
beam-pointing instabilities, light power fluctuations due to power supply or thermal fluctuations, are of a less fundamental nature and will
be disregarded here.
We consider the basic set-up of an ideal
Michelson interferometer where the measuring
beam is reflected by the backside of the cantilever of an AFM (fig. 1). For simplicity the
interferometer uses a collimated laser beam which
0304-3991/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
C.A.J. Putman et al. / Comparison between interferometric and OBD technique
1510
results in a change in the number of photons
incident at the detector of
A N = (2rr/A)tNto t Az,
(1)
B$
\
where A is the wavelength of the laser light and t
is the measuring time. The uncertainty in the
measurement of AN is determined by the statistical variation of the total number of photons
detected. The value of the equilibrium state has
been obtained over a long period of time in order
to minimize the uncertainty of that signal. For
A N << Ntot we can write for the signal-to-noise
ratio
Fig. 1. Set-up of a Michelson interferometer. The object beam
incident on the cantilever C interferes at the detector D E T
with the reference beam from the mirro M. BS is a beam
splitter.
is completely reflected by the cantilever. In practice the laser beam is focused on the cantilever
since the size of the reflecting surface of the
cantilever is usually smaller than the collimated
laser beam. This is, however, not essential for our
analysis.
If the photon-emission rate of the laser is
Ntot/S the detector will detect between 0 and Ntot
photons (per second), when the cantilever position varies over a distance of X/2 around its
equilibrium position. The sensitivity of the interferometer is highest when the slope of the curve
in fig. 2 is maximal, e.g. at point Q (operation at
quadrature). A small cantilever displacement A z
SNRin f = A N / ( tNtot/2)1/2
= (87"r2tNtot) ]/2 A z / A ,
(2)
where it is assumed that an ideal photon-counting
detector with a quantum efficiency of 1 and negligible dark-count rate is used. Adaptation of the
formula for realistic detectors is straightforward
but not necessary for a comparison with the optical beam deflection method.
3. The sensitivity of an ideal optical beam deflection system
In analyzing the fundamental sensitivity of the
optical beam deflection method it is clarifying to
Ntot
to
c
o
l-
x~
,[
z
Fig. 2. Response of the interferometer as a function of the tip
position, z. At quadrature, point Q, a change in cantilever
position Az, causes a change in the amount of photons
incident on the detector, AN.
Fig. 3. Collimated laser beam (diameter D) incident on the
cantilever (length l). The reflected light is focused by a lens
(focal length f ) onto the position-sensitive detector (PSD).
C.A.J. Putman et aL / Comparison between interferometric and OBD technique
consider first the set-up of fig. 3. A collimated
Gaussian laser beam of diameter D is totally
reflected by the cantilever of length 1 and focused
by a lens (focal length f ) on the center of a split
detector. For simplicity it is assumed that the
angle between laser beam and normal to the
cantilever is small. In the focal plane of the lens
the irradiance distribution is still Gaussian with a
waist diameter given by
d o = 4Af/~rO.
(4)
where it is assumed that the cantilever is displaced as a solid body and the separation of the
two detectors is much smaller than d o. The displacement of the laser spot is measured by subtracting the signals of the two detectors of the
split detector. Assuming ideal photon counters as
detectors we can write for this signal
AN = tNtot x 2 × ( 2 r r ) I / 2 ( D / I ) ( A z / A ) .
(5)
The numerical factor is due to the Gaussian
shape of the irradiance distribution at the split
detector. The uncertainty in the difference signal
is again determined by the statistical variation in
the number of photons that are detected by the
two detectors during the measurement. Assuming
a Poisson distribution for the photon-emission
rate, we have for the signal-to-noise ratio
SNRobdl = A N / ( t N t o t / 2 + tNtot/2 ) 1/2
= (87rtNtot)l/2(n/l)(Az/a).
LASER
(3)
A cantilever displacement Az results in a shift of
the laser spot at the split detector As
As = 2 f A z / l ,
1511
(6)
Note that the expression is not dependent on the
focal length of the lens. The equation is in agreement with the proportionality relation stated by
Meyer and Amer [10]. The relation suggests that
D / l should be made as large as possible. This
ratio is limited to a value of about one by the
requirement in our analysis that all the laser light
is reflected by the cantilever. If the diameter of
the laser D exceeds the length of the cantilever,
light will be lost and the irradiation pattern in the
Fig. 4. Laser b e a m (from a diode laser) focused by a lens at
the cantilever. T h e reflected beam is detected by the PSD.
Distance between cantilever and PSD is X.
focal plane of the lens will no longer be Gaussian.
Both effects can be shown to decrease the signalto-noise ratio. In the optimal case for which D = l,
the expression for the signal-to-noise ratio of the
OBD method reduces to
SNRobdl =
(8"trtNtot) 1/2
AZ/A.
(7)
If we compare this with the expression obtained
for the Michelson interferometer, eq. (2), we obtain the remarkable result that both methods
yield essentially the same sensitivity.
The set-up discussed above cannot be used in
a real AFM because the length of most cantilevers is so small that due to diffraction a collimated laser beam of this size cannot be constructed. Therefore, we consider the situation of
fig. 4 where the Gaussian laser beam is focused
onto the cantilever. Let the waist at the cantilever
be D O. After reflection the diameter of the beam
at a distance x from the cantilever is given by [13]
D ( x ) = Do[1
The ideal
distance x
the beam.
placement
+
( 4 , h x / ~ D 2 ) 2 ] 1/2
(8)
split detector is now positioned at a
from the cantilever and centered in
The difference signal for a small disA z of the cantilever is given by
AN = 8(2/Tr)l/2tNtot(x/D(x))(Az/l).
(9)
C.A.J. Putman et al. / Comparison between interferometric and OBD technique
1512
This yields a signal-to-noise ratio of
SNRoba2 = 8( tNto t X 2/'n') 1/2
x
× D0[1 +
(4Ax/TrD~)2]1/2Az/l"
(10)
We see that the signal-to-noise ratio in this situation is maximal when x >>
In that case
the formula is exactly the same as in the previous
case, eq. (6). That means that when
is close
to the maximal allowed value of one, also this
configuration of the optical b e a m deflection
method is just as sensitive as the interferometric
method.
7rD~/4,~.
Doll
4. Discussion
We have found that the sensitivity of a Michelson interferometer and that of two versions of the
optical beam deflection method for the measurement of cantilever displacements are essentially
the same. This suggests that the fundamental
physical principles underlying both methods are
also the same. In the following discussion we will
show this by noticing that a b e a m displacement
method can be viewed as an interferometric technique as well.
Suppose the collimated laser b e a m in the
scheme of fig. 3 is replaced by two parallel collimated beams originating from the same coherent
light source. The first b e a m is reflected by the tip
of the cantilever, the second b e a m from a position close to the base of the cantilever. In the
focal plane of the lens the two beams completely
overlap and form an interference pattern at the
detector surface. A displacement of the tip of the
cantilever results in a shift of this interference
pattern. By a m e a s u r e m e n t of this shift we have
constructed an interferometric method to detect
the cantilever displacement. In fact, the interferometer described by Den Boef [7] is very similar
to this. The sensitivity of this interferometric
method is equivalent to any other ideal interferometer such as the Michelson interferometer described in fig. 1. The sensitivity of an interferome-
ter is optimal when two conditions are fulfilled.
Firstly, the two beams must overlap completely.
In that case all information on the phase difference between the two beams iscontained in the
interference pattern. In the hypothetical set-up
described above this is achieved in the focal plane
of the lens. Secondly, the change in phase difference between the two beams due to a displacement of the cantilever should be maximal. This
condition is also fulfilled since one b e a m is at the
tip of the cantilever and the other at the base.
In the OBD set-up described in fig. 3, the
outer parts of the laser b e a m can be viewed as
the two beams mentioned above. The interference pattern in the focal plane of the lens is now
Gaussian and an optimal m e a s u r e m e n t of a shift
of this pattern can be done with a simple split
detector.
Generalizing these ideas, we come to the following rules that should be followed for designing
an optimal O B D set-up:
• Light reflected by different points of the cantilever should completely interfere at the detector. This is achieved when the detector plane is
the Fourier transform of the cantilever plane.
This can be obtained either by using a lens as
we did in the first example, or by placing the
detector in the far-field region of the cantilever. Essentially this was done when we concluded that x should be larger than 7rD~/4A
for an optimal signal-to-noise ratio in our second configuration.
• The total length of the cantilever should be
illuminated. This will maximize the differences
in phase shift introduced by a displacement of
the cantilever for different parts of the beam.
This corresponds to the conditions D = l and
D O= l in the two cases considered.
• All the available light should be used in order
to minimize shot noise.
It has been the aim of this study to compare
the fundamental limitations of the O B D with
interferometric methods. In practice, other considerations such as available space, beam-pointing stability of the laser, electronic noise, mechanical vibrations, etc. will be equally important
design criteria. One of the main problems with
C.A.J. Putman et al. / Comparison between interferometric and OBD technique
inteferometric methods is the occurrence of unwanted phase differences between measuring
beam and reference beam. This has been greatly
reduced by the fiber interferometer of Rugar et
al. [9] and the design of Den Boef [7]. In the
OBD this is automatically fulfilled. Considered as
an interferometer, the interfering beams have
essentially the same optical path except in the
near-field region of the cantilever.
The reported sensitivities of the various configurations using interferometry and the set-up
using optical beam deflection are as follows. For
interferometry-based techniques the values are:
1.7 x 10 -3 ,~/(Hz) 1/2 (homodyne) [5], 8 x 10 -4
A/(Hz) 1/2 (common path) [7], 7 x 10 -5 , ~ /
(Hz) 1/2 (polarized differential) [8] and 5.5 x 10 -4
,~/(Hz) 1/2 (fiber) [9]. Meyer and Amer [10] reported for the optical beam deflection technique
a minimum dectable displacement of 1 x 10 -3
,~/(Hz) 1/2. Thus it can be concluded that in
practical situations optical beam deflection and
interferometry have about the same sensitivity.
5. Conclusions
The theoretical analysis presented here shows
that a properly designed optical beam deflection
configuration is as sensitive as any interferometric method. Because of its simple construction
and the absence of a reference beam, optical
beam deflection is a very attractive measuring
method for an atomic force microscope.
1513
Acknowledgements
We thank Dr. B. B61ger for a critical reading
of this paper. This work was financially supported
by the Netherlands Organization for Scientific
Research (NWO).
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